Question

The number of natural numbers less than ${10}^7,$ whose sum of digits is equal to $6,$ is

• A. $1024$
• B. $924$
• C. $824$
• D. $720$

Answer 1

The correct option is B $924$

Let the digits of the numbers be $x_1,x_2,x_3,...,x_7$
so, $x_1+x_2+x_3+\cdots +x_7=6$
where $0\le x_i\le 6,i=1,2,3,...,7$

Using Multinomial theorem
$(1+x+x^2+\cdots +x^6)(1+x+x^2+\cdots +x^6)...\text{7 times}...(1+x+x^2+\cdots +x^6)\to \text{coefficient of}{x}^6$
$\Rightarrow (1+x+x^2+\cdots +x^6{)}^7\to x^6$
$\Rightarrow {\left(\frac{1-x^7}{1-x}\right)}^7\to x^6$
$\Rightarrow (1-x^7{)}^7(1-x{)}^{-7}\to x^6$
$\Rightarrow 1\cdot (1-x{)}^{-7}\to x^6$
$\therefore (1-x{)}^{-r}=\sum _{n=1}^{\infty }{}^{n+r-1}{C}_n{x}^n$
$\Rightarrow (1-x{)}^{-7}=\sum _{n=1}^{\infty }{}^{n+6}{C}_n{x}^n$
So the coefficient of $x^6\to {}^{6+6}{C}_6={}^{12}{C}_6=924$